Question

If $\left|x\right|<\frac{1}{2}$, the coefficient of $x^4$ in the expansion of $\frac{1}{(1+2x)(1-x^2)}$ is

• A. $1$
• B. $22$
• C. $21$
• D. none of these

Answer 1

The correct option is C $21$

$(1+2x{)}^{-1}(1-x^2{)}^{-1}$
$=(1-2x+4x^2-8x^3+16x^4+...\infty )(1+x^2+x^4+x^6+...\infty )$
Hence,from the above expansion the coefficient of $x^4$ will be
$=1+4\left(1\right)+16\left(1\right)$
$=1+4+16$
$=21$